What is a PRP ?
A PRP is a probable prime number, a number that nobody knows how to prove
or disprove its primality.
Here are some criterions that can be used to recognize these numbers
:
- N does not have any prime factor up to 2^32.
- N cannot be trivially written into a product.
- Take 3 square-free consecutive bases such that (b1/N)=(b2/N)=(b3/N)= -1. N
should be such that b1^((N-1)/2)=b2^((N-1)/2)=b3^((N-1)/2)= -1 (mod N).
- Take a prime base such that (b/N)= -1 and b<>b1<>b2<>b3.
N should pass the strong test in base b.
- or more (if someone has got an idea) !!!
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Rank |
Probable prime |
Digits (> 50000) |
Who ? |
When ? |
9751 | 19^58458+18 | 74754 | Lelio R Paula | 01/2015 |
9752 | 12^69265-35 | 74750 | Lelio R Paula | 02/2015 |
9753 | (2^248291+830)/2 | 74743 | Stefano Morozzi | 01/2020 |
9755 | 11^71771-48000 | 74742 | Enzo Doria | 12/2012 |
9754 | (7*10^74741+197)/3 | 74742 | Robert Price | 07/2017 |
9756 | (2^248282-5)/50129 | 74736 | Lelio R Paula | 06/2012 |
9757 | 2^248252-110105 | 74732 | Enzo Doria | 04/2014 |
9758 | (7*10^74729+71)/3 | 74730 | Robert Price | 03/2016 |
9759 | 3^156614-50 | 74724 | Lelio R Paula | 01/2015 |
9760 | (7*2^248213+1)/135 | 74719 | Lelio R Paula | 07/2012 |
9761 | (2^248193+870)/2 | 74714 | Stefano Morozzi | 01/2020 |
9762 | (2^248167+1178)/2 | 74706 | Stefano Morozzi | 01/2020 |
9763 | (2^248173+78557)/767 | 74705 | Lelio R Paula | 07/2012 |
9764 | (2^248145+3866)/2 | 74699 | Stefano Morozzi | 01/2020 |
9765 | 125^35622+124 | 74697 | Norbert Schneider | 02/2015 |
9766 | (2^248114+994)/2 | 74690 | Stefano Morozzi | 01/2020 |
9767 | (19*2^248099-1)/29 | 74686 | Lelio R Paula | 01/2014 |
9768 | 20780^3927+3927^20780 | 74685 | Hans Havermann | 12/2019 |
9769 | (86*10^74676-41)/9 | 74677 | Robert Price | 11/2015 |
9770 | (2^248065+3650)/2 | 74675 | Stefano Morozzi | 01/2020 |
9771 | (p(15711)#-4)/2 | 74672 | Robert Price | 09/2017 |
9772 | 10^74670+31 | 74671 | Robert Price | 01/2011 |
9773 | 10^74670-122463 | 74670 | Patrick De Geest | 05/2013 |
9774 | 2^248036-69 | 74667 | Lelio R Paula | 03/2012 |
9775 | 11^71698-6000 | 74666 | Enzo Doria | 12/2012 |
9776 | (2^248032+406)/2 | 74665 | Stefano Morozzi | 01/2020 |
9777 | 58*(10^(3*24887)-1)/111-1 | 74661 | Robert Price | 10/2014 |
9778 | 14^65139-45 | 74658 | Lelio R Paula | 01/2015 |
9779 | 17967^14288+14288^17967 | 74653 | Hans Havermann | 11/2019 |
9780 | 6^95921-133 | 74642 | Lelio R Paula | 01/2015 |
9781 | 11^71675+78 | 74642 | Lelio R Paula | 02/2015 |
9782 | 17875^14986+14986^17875 | 74641 | Hans Havermann | 12/2019 |
9783 | (2^247931+1458)/2 | 74635 | Stefano Morozzi | 01/2020 |
9784 | (166*10^74624-7)/3 | 74626 | Robert Price | 11/2020 |
9785 | 18^59449-91 | 74625 | Lelio R Paula | 01/2015 |
9786 | (12696^18181-1)/12695 | 74605 | Serge Batalov | 08/2020 |
9787 | 6^95864-16043 | 74597 | Enzo Doria | 02/2013 |
9788 | 5*7^88261+6 | 74590 | Lelio R Paula | 10/2014 |
9789 | 9*5^106710+4 | 74589 | Lelio R Paula | 10/2014 |
9790 | (777*2^247788+1)/(11*754121) | 74588 | Jens K. Andersen | 09/2007 |
9791 | 21157^3352+3352^21157 | 74585 | Hans Havermann | 12/2019 |
9792 | (265*10^74579+17)/3 | 74581 | Robert Price | 05/2016 |
9793 | 17798^15489+15489^17798 | 74575 | Hans Havermann | 11/2019 |
9794 | (2^247759-1017)/3318311717 | 74574 | Lelio R Paula | 07/2012 |
9795 | 5^106682-24 | 74568 | Lelio R Paula | 08/2014 |
9796 | (2^247686+1450)/2 | 74561 | Stefano Morozzi | 01/2020 |
9797 | 474^27863+27863^474 | 74556 | Norbert Schneider | 01/2017 |
9798 | (14*10^74540+79)/3 | 74541 | Robert Price | 11/2017 |
9799 | (2^247582+1630)/2 | 74530 | Stefano Morozzi | 01/2020 |
9800 | (35365^16384+1)/2 | 74524 | Lelio R Paula | 11/2011 |
9801 | (1705^23059-1)/1704 | 74518 | Serge Batalov | 08/2020 |
9802 | 3^156161+2660 | 74508 | Lelio R Paula | 01/2015 |
9804 | 2^247474-58695 | 74498 | Lelio R Paula | 05/2013 |
9803 | (2^247476+886)/2 | 74498 | Stefano Morozzi | 01/2020 |
9805 | 11^71519-90 | 74480 | Enzo Doria | 07/2012 |
9806 | (19*2^247419-1)/23202983 | 74475 | Lelio R Paula | 01/2014 |
9808 | (2^247394-7)/3 | 74473 | Lelio R Paula | 06/2012 |
9807 | 18^59328+17 | 74473 | Lelio R Paula | 01/2015 |
9809 | 2^247379-181 | 74469 | Lelio R Paula | 03/2012 |
9810 | (15*2^247370-1)/479 | 74465 | Lelio R Paula | 01/2014 |
9811 | 18860^8877+8877^18860 | 74465 | Hans Havermann | 12/2019 |
9812 | 17355^17564-17564^17355 | 74462 | Norbert Schneider | 11/2019 |
9813 | 2^247344+2047 | 74458 | Lelio R Paula | 05/2013 |
9814 | 2^247311-1017 | 74449 | Lelio R Paula | 05/2013 |
9815 | (2^247288+1938)/2 | 74441 | Stefano Morozzi | 01/2020 |
9816 | 2^247280+7 | 74439 | Lelio R Paula | 03/2012 |
9817 | (68*10^74434+13)/9 | 74435 | Ray Chandler | 11/2010 |
9818 | 3^156004-62 | 74433 | Lelio R Paula | 01/2015 |
9819 | 21865^2536+2536^21865 | 74432 | Hans Havermann | 11/2019 |
9820 | (2^247223+1050)/2 | 74422 | Stefano Morozzi | 01/2020 |
9821 | 36095!2-64 | 74416 | Robert Price | 06/2015 |
9822 | 67699!4+16 | 74409 | Robert Price | 09/2019 |
9823 | 2^247165-3 | 74405 | M. Frind & P. Underwood | 07/2002 |
9824 | 2^247150+32769 | 74400 | Lelio R Paula | 05/2013 |
9825 | (2^247130+3858)/2 | 74394 | Stefano Morozzi | 01/2020 |
9826 | (2^247055+2426)/2 | 74371 | Stefano Morozzi | 01/2020 |
9827 | 12^68911+73 | 74368 | Lelio R Paula | 01/2015 |
9828 | 2^246999+74903 | 74355 | Enzo Doria | 04/2016 |
9829 | 2^246989-110643 | 74352 | Enzo Doria | 05/2014 |
9830 | 10^74350+279439 | 74351 | Patrick De Geest | 07/2017 |
9833 | (10^74350+11)/3 | 74350 | Robert Price | 11/2014 |
9831 | 10^74350-400000001 | 74350 | Robert Price | 07/2017 |
9832 | 10^74350-716241 | 74350 | Patrick De Geest | 07/2017 |
9834 | 67647!4-512 | 74346 | Robert Price | 11/2019 |
9835 | 10^74335+13659 | 74336 | Patrick De Geest | 05/2013 |
9836 | 10^74335-53 | 74335 | Robert Price | 05/2011 |
9837 | 23360^1521+1521^23360 | 74335 | Hans Havermann | 11/2019 |
9838 | (2^246914+3418)/2 | 74329 | Stefano Morozzi | 01/2020 |
9839 | 2^246906-1215 | 74327 | Lelio R Paula | 05/2013 |
9840 | (2^246879+1494)/2 | 74318 | Stefano Morozzi | 01/2020 |
9841 | 2^246848+111 | 74309 | Lelio R Paula | 03/2012 |
9842 | (2^246842+2490)/2 | 74307 | Stefano Morozzi | 01/2020 |
9844 | 6^95490-18443 | 74306 | Enzo Doria | 03/2013 |
9843 | (2^246840+2902)/2 | 74306 | Stefano Morozzi | 01/2020 |
9845 | 18^59188-2369 | 74298 | Enzo Doria | 03/2013 |
9846 | (1363*2^246767+1)/(3*5*7*13^2*743) | 74281 | Jens K. Andersen | 09/2007 |
9847 | 2^246744+40755 | 74278 | Lelio R Paula | 05/2013 |
9848 | (29*10^74276+91)/3 | 74277 | Robert Price | 03/2016 |
9849 | 17222^17533-17533^17222 | 74272 | Norbert Schneider | 11/2019 |
9850 | (2^246720+3606)/2 | 74270 | Stefano Morozzi | 01/2020 |
9851 | 6^95441+163 | 74268 | Lelio R Paula | 01/2015 |
9852 | (40734^16111-1)/40733 | 74267 | Anonymous | 11/2020 |
9853 | (14235^17881-1)/14234 | 74263 | Serge Batalov | 08/2020 |
9854 | V(390863) | 74258 | Serge Batalov | 03/2015 |
9855 | 10^74254+91 | 74255 | Robert Price | 03/2011 |
9857 | (19*10^74253-1)/9 | 74254 | Ray Chandler | 02/2012 |
9856 | 10^74254-130569 | 74254 | Patrick De Geest | 05/2013 |
9858 | 771^25718+25718^771 | 74250 | Norbert Schneider | 10/2016 |
9859 | 2^246618+427 | 74240 | Henri Lifchitz | 02/2005 |
9860 | 88*10^74234+3 | 74236 | Robert Price | 11/2016 |
9861 | 8*10^74234-49 | 74235 | Robert Price | 04/2016 |
9862 | (19648^17291-1)/19647 | 74232 | Serge Batalov | 08/2020 |
9863 | 2^246578+12285 | 74228 | Lelio R Paula | 05/2013 |
9864 | (3^155569-5)/2 | 74225 | Lelio R Paula | 06/2012 |
9865 | 13^66628+46 | 74220 | Lelio R Paula | 01/2015 |
9866 | (2^246534+3034)/2 | 74214 | Stefano Morozzi | 01/2020 |
9867 | (2^246486+798)/2 | 74200 | Stefano Morozzi | 01/2020 |
9868 | 2^246476-17295 | 74197 | Lelio R Paula | 05/2013 |
9869 | (2^246465+446)/2 | 74194 | Stefano Morozzi | 01/2020 |
9870 | (61*10^74189-1)/3 | 74191 | Robert Price | 11/2017 |
9871 | (15185*2^246432-1)/13 | 74187 | Lelio R Paula | 04/2015 |
9872 | 5*2^246413-3 | 74179 | Lelio R Paula | 05/2014 |
9873 | 2^246386-28635 | 74170 | Lelio R Paula | 05/2013 |
9874 | (2^246371+3050)/2 | 74165 | Stefano Morozzi | 01/2020 |
9875 | 2^246334-111245 | 74154 | Enzo Doria | 05/2014 |
9876 | 3*9^77708+7^77708-2*4^77708-3 | 74153 | Mike Oakes | 12/2010 |
9877 | 15^63045+82 | 74147 | Lelio R Paula | 01/2015 |
9878 | 19255!+19255#+1 | 74140 | D. Gazzoni Filho & Lucas Teixeira | 06/2005 |
9879 | (199^32251-1)/198 | 74139 | Serge Batalov | 08/2020 |
9880 | (2^246277+2114)/2 | 74137 | Stefano Morozzi | 01/2020 |
9881 | 2^246268-110507 | 74135 | Enzo Doria | 05/2014 |
9882 | 3^155328+2660 | 74111 | Lelio R Paula | 01/2015 |
9884 | 2^246148-2786249 | 74098 | Enzo Doria | 03/2016 |
9883 | 19740^5671+5671^19740 | 74098 | Hans Havermann | 01/2020 |
9885 | (2^246135+3386)/2 | 74094 | Stefano Morozzi | 01/2020 |
9886 | (16*10^74092-19)/3 | 74093 | Robert Price | 03/2016 |
9887 | 2^246115-110767 | 74088 | Enzo Doria | 05/2014 |
9888 | 7^87666-68 | 74087 | Lelio R Paula | 08/2014 |
9889 | (2^246091+1938)/2 | 74081 | Stefano Morozzi | 01/2020 |
9890 | 10^74079+1145141919 | 74080 | Marisa Kirisame | 03/2015 |
9891 | 10^74079+6219 | 74080 | Patrick De Geest | 03/2016 |
9892 | 10^74079-282707 | 74079 | Patrick De Geest | 03/2016 |
9893 | 129^35098+2 | 74078 | Lelio R Paula | 05/2013 |
9894 | 22193^2176+2176^22193 | 74073 | Hans Havermann | 12/2019 |
9895 | 17703^15278+15278^17703 | 74071 | Hans Havermann | 12/2019 |
9896 | 2^246025-112123 | 74061 | Enzo Doria | 06/2014 |
9897 | (67*10^74045+17)/3 | 74047 | Robert Price | 02/2017 |
9898 | (2^245944+2202)/2 | 74037 | Stefano Morozzi | 01/2020 |
9899 | (2^245885+1446)/2 | 74019 | Stefano Morozzi | 01/2020 |
9900 | (13769^17881-1)/13768 | 74004 | Serge Batalov | 08/2020 |
9901 | 2^245823+66831 | 74001 | Marek Hubal | 02/2017 |
9902 | 5^105869+42 | 74000 | Lelio R Paula | 08/2014 |
9903 | (21*2^245806+1)/634555 | 73991 | Lelio R Paula | 08/2012 |
9904 | (22699*2^245762+1)/119 | 73985 | Lelio R Paula | 06/2014 |
9905 | (3^155048-7)/262 | 73975 | Lelio R Paula | 06/2012 |
9906 | 9998*2^245696+3 | 73966 | Donovan Johnson | 10/2004 |
9907 | 13^66400-80 | 73966 | Lelio R Paula | 01/2015 |
9909 | (2^245704+9)/25 | 73963 | Lelio R Paula | 07/2012 |
9910 | 11^71023-7170 | 73963 | Enzo Doria | 01/2013 |
9908 | 3*10^73962+13 | 73963 | Robert Price | 08/2017 |
9911 | (64758^15373-1)/64757 | 73960 | Anonymous | 11/2020 |
9912 | (31*2^245660-1)/114195 | 73948 | Lelio R Paula | 01/2014 |
9913 | 2^245629-2797813 | 73942 | Eric Chen | 03/2016 |
9914 | 2^245557-153 | 73921 | Lelio R Paula | 03/2012 |
9915 | (77*10^73917-59)/9 | 73918 | Robert Price | 11/2017 |
9916 | 15010^17699+17699^15010 | 73918 | Norbert Schneider | 11/2019 |
9917 | (12975^17971-1)/12974 | 73913 | Serge Batalov | 08/2020 |
9918 | 2^245525+1869881 | 73911 | Enzo Doria | 06/2016 |
9919 | 2^245522+159 | 73910 | Lelio R Paula | 03/2012 |
9920 | (2^245507+1986)/2 | 73905 | Stefano Morozzi | 01/2020 |
9921 | (13*10^73891-43)/3 | 73892 | Robert Price | 08/2017 |
9922 | 19^57783-56 | 73891 | Lelio R Paula | 01/2015 |
9923 | 7*10^73888+79 | 73889 | Robert Price | 02/2017 |
9924 | (2^245449+942)/2 | 73888 | Stefano Morozzi | 01/2020 |
9925 | 6^94948+174281 | 73884 | Mike Oakes | 02/2011 |
9926 | (2^245431+3630)/2 | 73882 | Stefano Morozzi | 01/2020 |
9927 | 11^70939-4860 | 73876 | Enzo Doria | 01/2013 |
9928 | 18637^9196+9196^18637 | 73870 | Hans Havermann | 01/2020 |
9929 | 2^245372+54615 | 73865 | Lelio R Paula | 05/2013 |
9930 | 2^245362-143 | 73862 | Lelio R Paula | 03/2012 |
9931 | 7^87396+6^(87396-1) | 73859 | Robert Price | 03/2016 |
9932 | (22*10^73853-37)/3 | 73854 | Robert Price | 05/2017 |
9933 | 5*10^73851-69 | 73852 | Robert Price | 02/2021 |
9934 | 5^105646-14246 | 73844 | Enzo Doria | 02/2013 |
9936 | 2^245290-41 | 73840 | Lelio R Paula | 03/2012 |
9935 | 2^245290-94395 | 73840 | Lelio R Paula | 05/2013 |
9937 | 15770^17589+17589^15770 | 73836 | Norbert Schneider | 10/2019 |
9938 | 12352^18043+18043^12352 | 73828 | Norbert Schneider | 11/2019 |
9939 | (2^245240+3598)/2 | 73825 | Stefano Morozzi | 01/2020 |
9940 | (2^245242+498)/2 | 73825 | Stefano Morozzi | 01/2020 |
9941 | 8*5^105600-7 | 73813 | Lelio R Paula | 10/2014 |
9942 | 7*5^105573-4 | 73794 | Lelio R Paula | 10/2014 |
9943 | (2^245112+1122)/2 | 73786 | Stefano Morozzi | 01/2020 |
9944 | 20151^4570+4570^20151 | 73751 | Hans Havermann | 12/2019 |
9945 | (2^244968+1318)/2 | 73743 | Stefano Morozzi | 01/2020 |
9946 | 18^58742-4805 | 73738 | Enzo Doria | 03/2013 |
9947 | (7*10^73729+23)/3 | 73730 | Robert Price | 06/2020 |
9948 | 13^66179+66 | 73720 | Lelio R Paula | 01/2015 |
9949 | (2^244881+942)/2 | 73717 | Stefano Morozzi | 01/2020 |
9950 | (2^244883+1946)/2 | 73717 | Stefano Morozzi | 01/2020 |
9951 | (82*10^73714-1)/9 | 73715 | Robert Price | 12/2014 |
9952 | (2^244862+3682)/2 | 73711 | Stefano Morozzi | 01/2020 |
9953 | (29*2^244856-1)/186121 | 73706 | Lelio R Paula | 01/2014 |
9954 | (13*2^244840-1)/2799 | 73702 | Lelio R Paula | 01/2014 |
9955 | (13*10^73701+17)/3 | 73702 | Robert Price | 06/2015 |
9956 | 2339662057597*10^73677+9 | 73690 | Robert Price | 03/2021 |
9957 | 6*10^73686-29 | 73687 | Robert Price | 08/2020 |
9958 | (2^244787-1017)/28411 | 73684 | Lelio R Paula | 07/2012 |
9959 | 21213^2974+2974^21213 | 73680 | Hans Havermann | 01/2020 |
9960 | (3^154421-4)/23 | 73677 | Lelio R Paula | 06/2012 |
9961 | 14596^17691+17691^14596 | 73670 | Norbert Schneider | 11/2019 |
9962 | 8*7^87170+5 | 73669 | Lelio R Paula | 10/2014 |
9963 | (29*10^73659-113)/3 | 73660 | Robert Price | 11/2017 |
9964 | Phi(40100,-40100) | 73651 | Henri Lifchitz | 06/2013 |
9965 | (3883^20521-1)/3882 | 73650 | Serge Batalov | 08/2020 |
9966 | (4*10^73647-19)/3 | 73648 | Robert Price | 06/2017 |
9967 | 2^244618-2297 | 73638 | Yves Minet | 10/2009 |
9968 | 3*5^105344+8 | 73633 | Lelio R Paula | 10/2014 |
9969 | (2^244588+2562)/2 | 73629 | Stefano Morozzi | 01/2020 |
9970 | 5^105334-14096 | 73626 | Enzo Doria | 02/2013 |
9971 | (2^244562+1534)/2 | 73621 | Stefano Morozzi | 01/2020 |
9972 | (2^244584-7)/13237074243033 | 73614 | Lelio R Paula | 07/2012 |
9973 | 2^244465-110185 | 73592 | Enzo Doria | 05/2014 |
9974 | (2^244465+2934)/2 | 73591 | Stefano Morozzi | 01/2020 |
9975 | 2^244443+40755 | 73585 | Lelio R Paula | 05/2013 |
9976 | 23155^1506+1506^23155 | 73583 | Hans Havermann | 12/2019 |
9977 | 10^73579+987654321 | 73580 | Robert Price | 12/2019 |
9978 | 10^73579+124173 | 73580 | Patrick De Geest | 02/2020 |
9979 | 10^73579-176379 | 73579 | Patrick De Geest | 02/2020 |
9980 | 2^244404+3841 | 73573 | Yves Minet | 03/2011 |
9981 | 19770^5263+5263^19770 | 73569 | Hans Havermann | 12/2019 |
9982 | 9*10^73565-41 | 73566 | Robert Price | 08/2020 |
9983 | 2^244372-110627 | 73564 | Enzo Doria | 05/2014 |
9984 | 19778^5241+5241^19778 | 73563 | Hans Havermann | 12/2019 |
9985 | (2^244349+2186)/2 | 73557 | Stefano Morozzi | 01/2020 |
9986 | (2^244309+3710)/2 | 73545 | Stefano Morozzi | 01/2020 |
9987 | 18428^9783+9783^18428 | 73537 | Hans Havermann | 01/2020 |
9988 | (3*2^244255+1)/245 | 73527 | Lelio R Paula | 08/2012 |
9989 | (11*2^244245+1)/23 | 73525 | Lelio R Paula | 08/2012 |
9990 | 18140^11301+11301^18140 | 73524 | Hans Havermann | 12/2019 |
9991 | (15185*2^244187-1)/6111 | 73509 | Lelio R Paula | 04/2015 |
9992 | (9699^18439-1)/9698 | 73508 | Serge Batalov | 08/2020 |
9993 | (2^244152+2058)/2 | 73497 | Stefano Morozzi | 01/2020 |
9994 | (2^244109+1422)/2 | 73484 | Stefano Morozzi | 01/2020 |
9995 | 17063^17362+17362^17063 | 73477 | Norbert Schneider | 04/2019 |
9996 | (91^37507+1)/92 | 73476 | Paul Bourdelais | 08/2017 |
9997 | 18^58521+65 | 73460 | Lelio R Paula | 01/2015 |
9998 | 13^65940+36 | 73454 | Lelio R Paula | 01/2015 |
9999 | 19065^7112+7112^19065 | 73439 | Hans Havermann | 12/2019 |
10000 | (625^26267+1)/626 | 73437 | Richard Fischer | 12/2017 |